On The Eigenvalues And Moore-Penrose Inverses Of The Matrices Of Graphs

Let G be a simple undirected graph. The graph G is said to have forcing (respectively, admissible) reciprocal skew eigenvalue property if each eigenvalue A of skew-adjacency matrix S(G) and its reciprocal have the same multiplicity as eigenvalues of S(G) for all (respectively, some) orientations G of G.A graph is a threshold if it can be obtained from K1, by iterating the operations of adding in a new vertex which is connected to no other vertex (an isolated vertex) or adding in a new vertex connected to every other vertex (a cone vertex). An antiregular graph is a simple connected graph with at most two vertices of equal degree.In this thesis, we study the unicyclic graphs with forcing (respectively, admissible) reciprocal skew eigenvalue property, obtain the spectral properties and Smith normal forms of the distance matrices of antiregular graphs, and determine the Moore-Penrose inverses of some matrices of a connected threshold graph.This thesis consists of four chapters. In chapter one, we first introduce some basis concepts, notations and terminology. Then we point out the research developments in this area and briefly introduce the main results obtained in this paper.In the second chapter, the structure of the unicyclic graphs with forcing (or admissible) reciprocal skew eigenvalue property is determined. It is shown that the unicyclic graph with the property (ASR) but without the property (SSR) must have the property (R).In the third chapter, we give spectral properties of interlacing, main eigen-values and the inertial property of the distance matrix of any antiregular graph. At last, we obtain the Smith normal form of the distance matrice of any an-tiregular graph, which has unique nontrivial invariant.In the fourth chapter, the Moore-Penrose inverses of adjacency matrix, Randic matrix and Laplacian matrix of a threshold graph are determined. Re-sults for antiregular graphs follow as special cases. As an application, we obtain the expression for the resistance distance between two vertices in a threshold graph.